A topological base is a collection of open sets that can be used to generate all open sets in a topological space. It is a fundamental tool in topology that helps define the open sets and the topology of a space.
The two definitions of topological bases are the open subcollection definition and the neighborhood definition. The open subcollection definition states that a collection of open sets is a base if every open set in the topology can be expressed as a union of sets in the base. The neighborhood definition states that a collection of open sets is a base if every point in the space has a neighborhood in the base.
Yes, the two definitions of topological bases are equivalent. This means that a collection of open sets satisfies one definition if and only if it satisfies the other definition. In other words, a collection of open sets is a base if and only if every open set can be expressed as a union of sets in the base or if every point has a neighborhood in the base.
To prove that two definitions of topological bases are equivalent, you can show that they satisfy the same conditions. For example, you can show that a collection of open sets satisfies the open subcollection definition if and only if it satisfies the neighborhood definition.
Topological bases are important in topology because they help define the open sets and the topology of a space. They also allow for easier analysis and understanding of topological properties, such as connectedness and compactness. Bases also play a crucial role in the construction of topological spaces, such as quotient spaces and product spaces.